An advection-robust Hybrid High-Order method for the Oseen problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element $T$ of diameter $h_T$ contributes to the discretization error with an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition

Two-phase Discrete Fracture Matrix models with linear and nonlinear transmission conditions

International audienceThis work deals with two-phase Discrete Fracture Matrix models coupling the two-phase Darcy flow in the matrix domain to the two-phase Darcy flow in the network of fractures represented as co-dimension one surfaces. Two classes of such hybrid-dimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix-fracture interfaces. The linear transmission conditions include the cell-centred upwind approximation of the phase mobilities classicaly used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix-fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equi-dimensional model for which the fractures are represented as full-dimensional subdomains. The discretization focuses on Finite Volume cell-centred Two-Point Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix-fracture interfacial unknowns together with the non-linear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix-fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme

Numerical resolution of partial differential equations with variable coefficients

Cette thèse aborde différents aspects de la résolution numérique des Equations aux Dérivées Partielles.Le premier chapitre est consacré à l'étude de la méthode Mixed High-Order (MHO). Il s'agit d'une méthode mixte de dernière génération permettant d'obtenir des approximations d'ordre arbitraire sur maillages généraux. Le principal résultat obtenu est l'équivalence entre la méthode MHO et une méthode primale de type Hybrid High-Order (HHO).Dans le deuxième chapitre, nous appliquons la méthode MHO/HHO à des problèmes issus de la mécanique des fluides. Nous considérons d'abord le problème de Stokes, pour lequel nous obtenons une discrétisation d'ordre arbitraire inf-sup stable sur maillages généraux. Des estimations d'erreur optimales en normes d'énergie et L2 sont proposées. Ensuite, nous étudions l'extension au problème d'Oseen, pour lequel on propose une estimation d'erreur en norme d'énergie où on trace explicitement la dépendance du nombre de Péclet local.Dans le troisième chapitre, nous analysons la version hp de la méthode HHO pour le problème de Darcy. Le schéma proposé permet de traiter des maillages généraux ainsi que de faire varier le degré polynomial d'un élément à l'autre. La dépendance de l'anisotropie locale du coefficient de diffusion est tracée explicitement dans l'analyse d'erreur en normes d'énergie et L2.La thèse se clôture par une ouverture sur la réduction de problèmes de diffusion à coefficients variables. L'objectif consiste à comprendre l'impact du choix de la formulation (mixte ou primale) utilisée pour la projection sur l'espace réduit sur la qualité du modèle réduit.This Ph.D. thesis deals with different aspects of the numerical resolution of Partial Differential Equations.The first chapter focuses on the Mixed High-Order method (MHO). It is a last generation mixed scheme capable of arbitrary order approximations on general meshes. The main result of this chapter is the equivalence between the MHO method and a Hybrid High-Order (HHO) primal method.In the second chapter, we apply the MHO/HHO method to problems in fluid mechanics. We first address the Stokes problem, for which a novel inf-sup stable, arbitrary-order discretization on general meshes is obtained. Optimal error estimates in both energy- and L2-norms are proved. Next, an extension to the Oseen problem is considered, for which we prove an error estimate in the energy norm where the dependence on the local Péclet number is explicitly tracked.In the third chapter, we analyse a hp version of the HHO method applied to the Darcy problem. The resulting scheme enables the use of general meshes, as well as varying polynomial orders on each face.The dependence with respect to the local anisotropy of the diffusion coefficient is explicitly tracked in both the energy- and L2-norms error estimates.In the fourth and last chapter, we address a perspective topic linked to model order reduction of diffusion problems with a parametric dependence. Our goal is in this case to understand the impact of the choice of the variational formulation (primal or mixed) used for the projection on the reduced space on the quality of the reduced model

Résolution numérique d'équations aux dérivées partielles à coefficients variables

This Ph.D. thesis deals with different aspects of the numerical resolution of Partial Differential Equations.The first chapter focuses on the Mixed High-Order method (MHO). It is a last generation mixed scheme capable of arbitrary order approximations on general meshes. The main result of this chapter is the equivalence between the MHO method and a Hybrid High-Order (HHO) primal method.In the second chapter, we apply the MHO/HHO method to problems in fluid mechanics. We first address the Stokes problem, for which a novel inf-sup stable, arbitrary-order discretization on general meshes is obtained. Optimal error estimates in both energy- and L2-norms are proved. Next, an extension to the Oseen problem is considered, for which we prove an error estimate in the energy norm where the dependence on the local Péclet number is explicitly tracked.In the third chapter, we analyse a hp version of the HHO method applied to the Darcy problem. The resulting scheme enables the use of general meshes, as well as varying polynomial orders on each face.The dependence with respect to the local anisotropy of the diffusion coefficient is explicitly tracked in both the energy- and L2-norms error estimates.In the fourth and last chapter, we address a perspective topic linked to model order reduction of diffusion problems with a parametric dependence. Our goal is in this case to understand the impact of the choice of the variational formulation (primal or mixed) used for the projection on the reduced space on the quality of the reduced model.Cette thèse aborde différents aspects de la résolution numérique des Equations aux Dérivées Partielles.Le premier chapitre est consacré à l'étude de la méthode Mixed High-Order (MHO). Il s'agit d'une méthode mixte de dernière génération permettant d'obtenir des approximations d'ordre arbitraire sur maillages généraux. Le principal résultat obtenu est l'équivalence entre la méthode MHO et une méthode primale de type Hybrid High-Order (HHO).Dans le deuxième chapitre, nous appliquons la méthode MHO/HHO à des problèmes issus de la mécanique des fluides. Nous considérons d'abord le problème de Stokes, pour lequel nous obtenons une discrétisation d'ordre arbitraire inf-sup stable sur maillages généraux. Des estimations d'erreur optimales en normes d'énergie et L2 sont proposées. Ensuite, nous étudions l'extension au problème d'Oseen, pour lequel on propose une estimation d'erreur en norme d'énergie où on trace explicitement la dépendance du nombre de Péclet local.Dans le troisième chapitre, nous analysons la version hp de la méthode HHO pour le problème de Darcy. Le schéma proposé permet de traiter des maillages généraux ainsi que de faire varier le degré polynomial d'un élément à l'autre. La dépendance de l'anisotropie locale du coefficient de diffusion est tracée explicitement dans l'analyse d'erreur en normes d'énergie et L2.La thèse se clôture par une ouverture sur la réduction de problèmes de diffusion à coefficients variables. L'objectif consiste à comprendre l'impact du choix de la formulation (mixte ou primale) utilisée pour la projection sur l'espace réduit sur la qualité du modèle réduit

An optimal control framework for adaptive neural ODEs

In recent years, the notion of neural ODEs has connected deep learning with the field of ODEs and optimal control. In this setting, neural networks are defined as solutions of a given ODE which is solved witha certain time discretization. The learning task consists in finding the ODE parameters as the optimal values of a sampled loss minimization problem. In the limit of infinite time steps, and data samples, we obtain a notion of continuous formulation of the problem. The practical implementation involves two discretization errors: a sampling error, and a time-discretization error. In this work, we develop a general optimal control framework to analyse the interplay between the above two errors. We prove that to approximate the solution of the fully continuous problem at a certain accuracy, we not only need a minimal number of training samples, but we also need to solve the control problem on the sampled loss function with some minimal accuracy. The theoretical analysis allows us to develop rigorous adaptive schemes in time and sampling, and give rise to a notion of adaptive neural ODEs. The performance of the approach is illustrated in several numerical examples

A hp-Hybrid High-Order method for variable diffusion on general meshes

International audienceIn this work, we introduce and analyze a hp-Hybrid High-Order method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We formulate hp-convergence estimates for both the energy-and L2-norms of the error, which are the first results of this kind for Hybrid High-Order methods. The estimates are fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on its (local) anisotropy. The expected exponential convergence behaviour is numerically shown on a variety of meshes for both isotropic and strongly anisotropic diffusion problems

Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations

International audienceThis paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [D. Di Pietro, A. Ern, hal-00918482]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity omponent), the energy-norm of the velocity and the L2-norm of the pressure converge with order k+1, while the L2-norm of the velocity (super-)converges with order k+2. The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes

A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion

International audienceThis paper presents an extension of Discrete Fracture Matrix (DFM) models to com-positional two-phase Darcy flow accounting for phase transitions and Fickian diffusion. The hybrid-dimensional model is based on nonlinear transmission conditions at matrix fracture (mf) interfaces designed to be consistent with the physical processes. They account in particular for the saturation jump induced by the different rock types, for the Fickian diffusion in the fracture width, as well as for the thermodynamical equilibrium formulated by complementary constraints. The model is validated by numerical comparison with a reference equi-dimensional model using a TPFA approximation in space and a fully implicit Euler time integration. It is also compared with the usual approach based on an harmonic averaging of the transmissivities at mf interfaces combined with a two-point upwinding of the mobilities jumping over the mf interfaces. Our approach is shown to provide basically the same accuracy than the equi-dimensional model as opposed to the classical harmonic averaging approach which is shown to exhibit physical inconsistency. It is then applied to simulate the desaturation by suction at the interface between a fractured Callovo-Oxfordian argilite storage rock and a ventilation tunnel with data set provided by Andra

Hybrid Finite Volume discretization of two-phase Discrete Fracture Matrix models with nonlinear interface solver

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